Mar 6, 2009

[Note: Many of you may be already knowing why. But there is no harm in reading this article and checking out whether what I say is what you know :) ]You would think - 'whats so special about this question ? The speed of rotation of earth and the speed of revolution of the moon around earth are relative to each other in such a way that moon rises 50 minutes later everyday'. This however does not explain why '50 minutes' and why 'later' everyday and not earlier ? Another answer you may find in books or on the web is "Earth rotates around itself in 24 hours, and moon takes about 29.5 days for one revolution round the earth - so divide 24 by 29.5. That's right answer, but still it does not throw any light of 'why' and 'how' aspects of the original question.

If you think a bit hard, you will realize its not that hard to figure out the answer. 50 minutes is the average time by which moon rise is delayed everyday. Depending upon where you are located on earth, the time can be less or more than 50 minutes. Why so ? That's because the plane of revolution of earth around the sun and that of the moon around the earth are at different angles. Please see the diagram below (I have modified the original diagram that's available at wikipedia.org):

The moon's plane of revolution around the earth is tilted with respect to earth's plane of rotation (around itself) by about 29 degrees. Had the orbit of moon's revolution been co-planar with the plane of earth's rotation (in other word's had moon circled around the earth in earth's equatorial plane), the moon- rise and moon-set times would be same along all the places which fell on the same longitude. In addition, all the places on earth would have experienced the same delay (of 50 minutes) in moon-rise time everyday. But since the planes are oriented in the way they are, the daily 'delay' in the moon-rise time is different at different places on earth and they average out to be about 50 minutes.

Also, moon's orbit around the earth is a bit elliptical (and not circular) with perigee (point of elliptical orbit closest to center of the orbit) being 3,63,104 km and apogee (point farthest from the center) being 4,05,696 km. The average distance of moon from earth is about 384,400 km. Let's assume a circular moon orbit to keep our calculations simple.

Let's now turn to the answer to the core question of this article. Why 50 minutes later ? Mind you, earth completes one rotation in 24 hours. That is, 360 degrees in (24 X 60 = 1440) minutes. This means earth takes 4 minutes (divide 1440 by 360) to rotate through 1 degree. The direction of rotation is from west to east. Moon revolves around the earth in the same direction - from west to east. Moon takes about 29.5 days to travel one complete cycle around the earth. That is, it travels 360 degrees in 29.5 days - which comes to around 12.20 degrees per day (Which is about 0.0085 degrees per minute. Go figure out how)

Now refer to the following diagrams. They show moon's revolution around the earth as seen from earth's 'North' pole. When seen from north pole, the earth would appear to be rotating counter-clockwise. The moon also would appear to be moving around the earth in counter-clockwise direction. The diagrams show positions of moon with respect to earth on two consecutive days.

Lets say, on Day 1 at 6:00 pm, the city that you stay in, is at point A on earth when it sees moon rising at it's eastern horizon (so 6.00 pm is the moon-rise time). The earth (and hence your city) is traveling west to east, i.e., counter clockwise as seen from top (north pole). So is the moon. But since earth is spinning much faster than the moon, it passes past the moon swiftly enough to cause the moon to rise to it's east and set to it's west within span of approximately half day.

Consider next diagram which represent earth and moon positions exactly after 24 hours - Day 2 at 6:00 pm. At this instance, your city which has advanced through 360 degrees during past 24 hours would again find itself at point A.

But moon has advanced by approximately 12.2 degrees in counter-clockwise direction around the earth (we've already seen how). In order for your city to see moon rise at its eastern horizon, it has to travel an additional distance of 12.2 degrees to arrive at point B. We know earth rotates at about 1 degree in 4 minutes. So it would take an additional of approximately '12.2 X 4', which is 48.8 minutes to see the moon rise at its eastern horizon. Got it ? But we said its 50 minutes.

Consider this: When your city is at point-A on Day-2, the moon has already advanced by 12.2 degrees. No moon-rise for you at point-A. By the time your city advances another 12.2 degrees ( 48.8 minutes) to reach point-B, the moon will have advanced additionally by approximately 0.40 degrees (Remember, the moon is continuously advancing - about 0.0085 degrees a minute. So 0.0085 X 48.8 = 0.40 approximately). So after your city has arrived at point A, in order for it to see the moon rise, earth has to advance by a total of 12.2 + 0.40 degrees, which is almost 12.6 degrees. So in all earth will have to rotate for 12.6 X 4 = 50.4 minutes in addition to completing one rotation (24 hrs) so that your city can see it's moon rise on Day-2. And don't forget, the moon's orbit is elliptical and not circular. This has to be factored in to find out the exact delay in moon-rise time. But it averages out to around 50 minutes per day.

You have some really nice diagrams. I also see that you have identified something missed by many who quote the formula you show. This is the additional time needed to actually catch teh moon. I have been researching this same subject and wish to let you know that the formula you use is only an approximation that is about 4% off and that the calculation for the actual value is not much more difficult. In addition, the values you use are not as stated. The periods you use are sun based and are greater than the actual times around, as you think they are. I am completing a paper on the subject and if you want a copy I can send it. I am on Gmail at noskosteve.Regards, Steve N.

Ok, your explanation on the moon's advancement is the reason for that 50min extra stretch over 12hrs is clear. I'm having another question on the moon's invisible time. that is after it sets it rises again after just 12hrs. How could this happen, is the 5 or degree tilt of the plane in which moon travels play a role here? I need to think more to digest this..

Crystal clear and easy to understand. I've been fretting over that darned moon sliding in the opposite direction as the earth overtakes it. Once I got that then the difference in moonrise times got me confounded again. Now that I understand the math behind it I feel SO much better. Shukriya!

(ii) The addition, in the simulation (i), of the little further complexity arising due to the earth's rotation around the Sun.

(iii) The changes, in the simulation (ii), for the elliptic orbit of the moon, and then also of the earth.

(iv) The changes, in the simulation (iii), due to the ~5 degrees tilt between the ecliptic and the equatorial great circles. This feature will take the simulation into 3D (whereas (i) through (iii) have been strictly in 2D).

(v) Set a realistic initial condition for the positions of the nodes (raahu, ketu) and predict eclipses.

Not sure though that online Python interpreters could have been available about 8 years ago when you wrote this post.

[And to think, I go jobless in India because my profile doesn't fit job profiles, whereas every other commentator's does!! Imagine: one year = 10--20 lakhs, and calculate the loss (my loss) due to the Indian IT/IIT/COEP dumbness. [Someone or the other from the industry (or many of them) is (are) morally guilty, for sure.]]

I really enjoyed reading your article. I found this as an informative and interesting post, so i think it is very useful and knowledgeable. I would like to thank you for the effort you have made in writing this article.